340
F. Bordry et al.
Table 8.1 Multipole coefficients, field components and field lines for pure multipole magnets
Magnet type n Normal (B n = 0)
Skew (A n = 0)
Dipole
1 B x = 0
B y = B 1
B x = A 1
B y = 0
Quadrupole 2 B x = B 2 y
B y = B 2 x
B x = A 2 x
B y = − A 2 y
Sextupole
3 B x = B 3 2xy
B y = B 3 (x 2 − y 2 )
B x = A 3 (x 2 − y 2 )
B y = − A 3 2xy
Octupole
4 B x = B 4 (3x 2 y − y 3 )
B y = B 4 (x 3 − 3xy 2 )
B x = A 4 (x 3 − 3xy 2 )
B y = − A 4 (3x 2 y − y 3 )
quadrupole (n = 2, linear field), to higher order multipoles such as the sextupole
(n = 3, quadratic field profile), octupole (n = 4, cubic field profile), and so on.
Besides its compact form, the complex notation is useful because there is a direct
relation between multipoles (order and strength) and beam properties. This is why
accelerator magnets are often characterised using their harmonic content in terms of
the B n and A n coefficients of the field expansion. Indeed, the pure multipolar fields
discussed so far can only be approximated to a suitable degree in real magnets. The
field generated by a magnet contains then all multipoles, normal and skew, i.e. a
dense harmonic spectrum. Symmetries cause cancellation effects, resulting in low
(ideally zero) non-allowed multipoles when compared to the multipoles allowed by
the magnet symmetry. Selected multipoles can be further reduced by using design
features such as optimization of the coil and iron geometry, or corrections such as
passive and active magnetic shims.
We define the field quality of an accelerator magnet as the relative difference
between the field produced and the ideal field distribution, usually a pure multipole,
in the region of interest for the beam, which is generally referred to as the good field
region. Depending on the shape of the good field region, it may be convenient to
quote field quality as an overall homogeneity (i.e. typically done for magnets
with a rectangular or elliptic aperture), or providing the spectrum of multipoles other
than the one corresponding to the main magnet function (ratio of A n and B n to the
main field strength, typically done for magnets with round bore). Whichever the
F. Bordry et al.
Table 8.1 Multipole coefficients, field components and field lines for pure multipole magnets
Magnet type n Normal (B n = 0)
Skew (A n = 0)
Dipole
1 B x = 0
B y = B 1
B x = A 1
B y = 0
Quadrupole 2 B x = B 2 y
B y = B 2 x
B x = A 2 x
B y = − A 2 y
Sextupole
3 B x = B 3 2xy
B y = B 3 (x 2 − y 2 )
B x = A 3 (x 2 − y 2 )
B y = − A 3 2xy
Octupole
4 B x = B 4 (3x 2 y − y 3 )
B y = B 4 (x 3 − 3xy 2 )
B x = A 4 (x 3 − 3xy 2 )
B y = − A 4 (3x 2 y − y 3 )
quadrupole (n = 2, linear field), to higher order multipoles such as the sextupole
(n = 3, quadratic field profile), octupole (n = 4, cubic field profile), and so on.
Besides its compact form, the complex notation is useful because there is a direct
relation between multipoles (order and strength) and beam properties. This is why
accelerator magnets are often characterised using their harmonic content in terms of
the B n and A n coefficients of the field expansion. Indeed, the pure multipolar fields
discussed so far can only be approximated to a suitable degree in real magnets. The
field generated by a magnet contains then all multipoles, normal and skew, i.e. a
dense harmonic spectrum. Symmetries cause cancellation effects, resulting in low
(ideally zero) non-allowed multipoles when compared to the multipoles allowed by
the magnet symmetry. Selected multipoles can be further reduced by using design
features such as optimization of the coil and iron geometry, or corrections such as
passive and active magnetic shims.
We define the field quality of an accelerator magnet as the relative difference
between the field produced and the ideal field distribution, usually a pure multipole,
in the region of interest for the beam, which is generally referred to as the good field
region. Depending on the shape of the good field region, it may be convenient to
quote field quality as an overall homogeneity (i.e. typically done for magnets
with a rectangular or elliptic aperture), or providing the spectrum of multipoles other
than the one corresponding to the main magnet function (ratio of A n and B n to the
main field strength, typically done for magnets with round bore). Whichever the